Here, we answer the questions affirmatively, by giving a natural extension that complements the submodularity ratio \(\gamma \) . We introduce the notion of an upper submodularity ratio \(\beta \) , that bounds how far a function deviates from submodularity when considering the effect of removing single elements. Intuitively, the parameter \(\beta \)  compares the loss by removing an entire set compared to the aggregate individual losses for each element. We call functions for which both ratios are bounded \((\gamma ,\beta )\) -weakly submodular. To motivate our definition, we show that the objective for subset selection problem considered by Das and Kempe [1]} is \((\gamma ,\beta )\) -weakly submodular with \(\beta = \gamma ^{-1}\)  (where \(\gamma \)  can be bounded using a sparse eigenvalue parameter \(\lambda _{\min }\)  of the covariance matrix as in [1]}).
